Quantum variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series

In this paper, we give the upper bounds on the variance for cubic moment of Hecke--Maass cusp forms and Eisenstein series respectively. For the cusp form case, the bound comes from a large sieve inequality for symmetric cubes. We also give some nontrivial bounds for higher moments of symmetric cube $L$-functions. For the Eisenstein series case, the upper bound comes from Lindel\"of-on-average type bounds for various $L$-functions. In particular, we establish the sharp upper bounds for the fourth moment of $\mathrm{GL}(2)\times \mathrm{GL}(2)$ $L$-functions and the eighth moment of $\mathrm{GL}(2)$ $L$-functions around special points $1/2+it_j$. Our proof is based on the work of Chandee and Li \cite{C-L20} about bounding the second moment of $\mathrm{GL}(4)\times \mathrm{GL}(2)$ $L$-functions.